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 A255211 a(n) = n*(n+1)*(7*n+2)/6. 8
 0, 3, 16, 46, 100, 185, 308, 476, 696, 975, 1320, 1738, 2236, 2821, 3500, 4280, 5168, 6171, 7296, 8550, 9940, 11473, 13156, 14996, 17000, 19175, 21528, 24066, 26796, 29725, 32860, 36208, 39776, 43571, 47600, 51870, 56388, 61161, 66196, 71500, 77080, 82943 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is the number of triangles of all sizes in a polyiamond of trapezoid shape with 3 sides of length n and the base of length 2*n. The number of triangular cells in the trapezoid is 3*n^2. This is half of a regular hexagon with side lengths n. The number of triangles oriented with their bases aligned with the base of the trapezoid is n*(n+1)*(2*n+1)/3 and the number oriented in the opposite direction is n^2*(n+1)/2. a(n) is the sum of these two. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Luce ETIENNE, Illustration a(1), a(2), a(3), a(4) and a(5) Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA G.f.: x*(3 + 4*x) / (1 - x)^4. - Colin Barker, Feb 17 2015 a(n) = Sum_{j=0..n-1} (n-j)*(3*n-2*j) = Sum_{j=1..n} j*(n+2*j) for n>0. a(n) = A000292(2*n) - A000292(n). - Bruno Berselli, Sep 22 2016 Sum_{n>=1} 1/a(n) = 21*HarmonicNumber(2/7)/5 - 6/5 = 0.44513027538601361333... . - Vaclav Kotesovec, Sep 22 2016 EXAMPLE From the second comment: a(1)= 2+1, a(2)= 10+6, a(3)= 28+18, a(4)= 60+40. MATHEMATICA Table[n (n + 1) (7 n + 2)/6, {n, 0, 50}] (* Bruno Berselli, Feb 17 2015 *) PROG (PARI) concat(0, Vec(x*(4*x+3)/(x-1)^4 + O(x^100))) \\ Colin Barker, Feb 17 2015 (PARI) vector(50, n, n--; n*(n+1)*(7*n+2)/6) \\ Bruno Berselli, Feb 17 2015 (Magma) [n*(n+1)*(7*n+2)/6 : n in [0..50]]; // Wesley Ivan Hurt, Apr 11 2021 CROSSREFS Partial sums of A022264. Cf. A000292, A000330, A006331, A002411, A033428, A212977. Sequence in context: A374296 A152618 A296947 * A172482 A362007 A212564 Adjacent sequences: A255208 A255209 A255210 * A255212 A255213 A255214 KEYWORD nonn,easy AUTHOR Luce ETIENNE, Feb 17 2015 EXTENSIONS Edited and extended by Bruno Berselli, Dec 01 2016 STATUS approved

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Last modified September 18 02:51 EDT 2024. Contains 375995 sequences. (Running on oeis4.)